In the Gaussian Movie DataBase (GMDB), reviewers like you rate movies with Gaussian scores. In particular, the

Question:

In the Gaussian Movie DataBase (GMDB), reviewers like you rate movies with Gaussian scores. In particular, the first person to rate a movie assigns a Gaussian (q, 1) review score Xi, where ro is the true "quality" of the movie. After n reviews, a movie's rating is Rn = ∑ni=1 Xi/n. Strangely enough, in the GMDB, reviewers are influenced by prior reviews; if after n - 1 reviews a movie is rated Rn-i = r, the nth review n will rate the movie Xn, a Gaussian (r, 1) random variable, conditionally independent of X1,....,Xn-1 given Rn-1 = r.
(a) Find E[Rn].
(b) Find the PDF fRn(r).
(c) Find Var [Rn].
(d) Interpret your results as n → ∞? Does the law of large numbers apply here?